Pinpointing Computation with Modular Queries in the Boolean Hierarchy

A modular query consists of asking how many (modulo m) of k strings belong to a fixed NP language. Modular queries provide a form of restricted access to an NP oracle. For each k and m, we consider the class of languages accepted by NP machines that ask a single modular query. Han and Thierauf [HT95] showed that these classes coincide with levels of the Boolean hierarchy when m is even or k ≤ 2m, and they determined the exact levels. Until now, the remaining case — odd m and large k — looked quite difficult. We pinpoint the level in the Boolean hierarchy for the remaining case; thus, these classes coincide with levels of the Boolean hierarchy for every k and m. In addition we characterize the classes obtained by using an NP(l) acceptor in place of an NP acceptor (NP(l) is the lth level of the Boolean hierarchy). As before, these all coincide with levels in the Boolean hierarchy. ∗Abteilung Theoretische Informatik, Universität Ulm, Oberer Eselsberg, 89069 Ulm, Germany. Email: [email protected]. On leave from SPIC Science Foundation, Madras, India. Research supported by the Alexander von Humboldt Fellowship. †Yale University, Dept. of Computer Science, P.O. Box 208285, New Haven, CT 06520-8285, USA. Email: [email protected]. Supported by grants CCR-8952528 and CCR-9415410 from the United States National Science Foundation. ‡Abteilung Theoretische Informatik, Universität Ulm, Oberer Eselsberg, 89069 Ulm, Germany. Email: [email protected].